Well, as far as I'm aware there are no predictions for the behavior of 2D objects at relativistic speeds. This would require knowledge of the mass and velocity of 2D objects, which we have none. It certainly can't be tested, even if there are predictions. There's really no good answer to this question, because a 2D space is only hypothetical.

Well, we can derive the Lorentz factor by reasoning about motion simply through 1 dimension: we align two ordinary 3-D coordinate systems so that their axes are all parallel, and set them in constant motion relative to one another so that the axes are always remaining parallel. Thus, an observer fixed to one system will measure an observer fixed in the other as being in linear (1-D) motion. Using the postulates of relativity, we can derive the Lorentz factor, and then compute the Lorentz transformation, for each axis independently. Time ends up needing a Lorentz transformation as well, just like the ordinary spatial axes. So we build up the "full" Lorentz transformations out of 1-D Lorentz transformations for each axis (including time).

"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker

At 7/18/2013 8:28:40 PM, AlbinoBunny wrote:... near the speed of light slows down time for the object, then what is the equivalent for crossing 2-dimensional space. Is there one?

Also, the object in motion doesn't measure it's own time as slowing down; "for the object", its own watch ticks "normally". It would only measure the time of other coordinate systems to be slower, while they would measure its watch to be ticking more slowly.

Not sure if I needed to say that, but there it is anyway.

"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker

At 7/18/2013 8:28:40 PM, AlbinoBunny wrote:... near the speed of light slows down time for the object, then what is the equivalent for crossing 2-dimensional space. Is there one?

Also, the object in motion doesn't measure it's own time as slowing down; "for the object", its own watch ticks "normally". It would only measure the time of other coordinate systems to be slower, while they would measure its watch to be ticking more slowly.

Not sure if I needed to say that, but there it is anyway.

I thought the outside world's time would be moving "quicker" relatively when compared to the object moving near light-speed?

At 7/18/2013 8:28:40 PM, AlbinoBunny wrote:... near the speed of light slows down time for the object, then what is the equivalent for crossing 2-dimensional space. Is there one?

Also, the object in motion doesn't measure it's own time as slowing down; "for the object", its own watch ticks "normally". It would only measure the time of other coordinate systems to be slower, while they would measure its watch to be ticking more slowly.

Not sure if I needed to say that, but there it is anyway.

I thought the outside world's time would be moving "quicker" relatively when compared to the object moving near light-speed?

An observer would, in its own reference frame, see its own watch ticking at a faster rate relative to clocks it measures to be moving at high velocities.

"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker

At 7/18/2013 8:28:40 PM, AlbinoBunny wrote:... near the speed of light slows down time for the object, then what is the equivalent for crossing 2-dimensional space. Is there one?

Also, the object in motion doesn't measure it's own time as slowing down; "for the object", its own watch ticks "normally". It would only measure the time of other coordinate systems to be slower, while they would measure its watch to be ticking more slowly.

Not sure if I needed to say that, but there it is anyway.

I thought the outside world's time would be moving "quicker" relatively when compared to the object moving near light-speed?

An observer would, in its own reference frame, see its own watch ticking at a faster rate relative to clocks it measures to be moving at high velocities.

At 7/18/2013 8:28:40 PM, AlbinoBunny wrote:... near the speed of light slows down time for the object, then what is the equivalent for crossing 2-dimensional space. Is there one?

Also, the object in motion doesn't measure it's own time as slowing down; "for the object", its own watch ticks "normally". It would only measure the time of other coordinate systems to be slower, while they would measure its watch to be ticking more slowly.

Not sure if I needed to say that, but there it is anyway.

I thought the outside world's time would be moving "quicker" relatively when compared to the object moving near light-speed?

An observer would, in its own reference frame, see its own watch ticking at a faster rate relative to clocks it measures to be moving at high velocities.

I think we agree... maybe. lol

So, as to your original question, we derive the equations of special relativity for each degree of freedom (basically, directions) one at a time anyway, so imagining "Einstein in Flatland" is no problem, really just something of a reduction of the full "4-D world" we end up with by reasoning about our spacetime.

"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker

At 7/21/2013 1:05:42 AM, AlbinoBunny wrote:I thought the outside world's time would be moving "quicker" relatively when compared to the object moving near light-speed?

All motion is relative to the observer, and no observer moves relative to itself. So quit imagining someone going fast. Imagine someone who is stopped, but who is observing someone else going fast. That way you won't get confused and think you'll see your own watch slow down.

At 7/18/2013 8:28:40 PM, AlbinoBunny wrote:... near the speed of light slows down time for the object, then what is the equivalent for crossing 2-dimensional space. Is there one?

The same. Special Relativity deals with objects moving at constant speed in a straight line, so as long as you travel in 1 or more dimensions your ship will experience space-time shrinkage and mass increase.